This is a repo for Stabilized Stokes Flow using FEniCSx by Caleb Munger.
This repo contains code for FEniCSx 0.9.0.0 that takes a black and white .png image as the inlet shape, and than computes the outlet shape (example below) post prcoessing the image to get the flow profile at the inlet and outlet is done using paraview 12.1. This code can be used to predict the shape of of a fluid at an outlet (assuming no mixing) for low Reynolds number exetrusion flow.
Overview
The solver uses stabilized Navier-Stokes flow to solve for the outlet profile. You will also need a black and white .png file of the inlet shape (see "Plus.png" as an example). The second required input is the flowrate ratio between the inner and outer flow profiles. A flowrate ratio of 1 means all of the flow will come from the inner countour, while 0 means all of the flow is in the outer contour.There is a third optional parameter which is the length of the mesh elements (the small the number, the more elements). It is recommend to start with a mesh length input of 0.05 (the width of the channel us non-dimensioanlized to be 1) and gradually decrease from there to a suitable resolution.
The examples generated below used a flowrate ratio 0.5
Inlet Example
This is an example of the inlet profile used
Inlet Profile Example
This is an example of the inlet profile streamtrace
Outlet Profile Example
This is an example of the outlet profile streamtrace
Duct Stokes Flow
The "DuctStokesFlow.py" is meant as a test file. Its inputs are where you want to name the gmsh mesh file (Ex: "DuctMesh"), the length of the mesh elements (this is the same as above, it is recommened to start with 0.1), and the length of the total domain.
The duct flow simulations stokes flow through a square cross-section duct, with no obstructions. If the length of the duct is long enough (4 should be enough) the outlet profile will be fully devolped channel flow.
If you want to use the "StokesChannelFlow.py" file, it is recommended you first run the "DuctStokesFlow.py" file to make sure you have everything set up correctly (note, there are extra packages you will need to install when running the "StokesChannelFlow.py"), but the duct flow is computational easier to run, and has a known output.
Stabilization Methods
The solver can be with 2 differnet element types to produce a stable output. P2-P1 (quadratic velocity and linear pressure) elements, also called "Taylor-Hood" elements are a stable mixed formulation element pair. The other set of elements that can be used are stabilized P1-P1 (linear velocity and pressure). Grad-Div stabilization is used to allow the use of P1-P1 elements. Grad-Div stabilization worked by ensuring the FE stiffness matrix is positive definite, ensuring a solution exists.
Stream Tracing
To streamtrace the solution a combined 4th and 5th order Runge-Kutta method (RK45) from Scipy is used. The streamtrace happens in 2 steps, a foward and reverse streamtrace. The foward streamtrace uses the mesh that is used to assign the boundary condition as starting seeds. After streamtracing these seeds, the output profile is found and expandly, and then reverse streamtrace seeds are placed in a grid. The velocity field is reversed, and after the reverse streamtrace, the seeds that end inside the inner contour are accepted and plotted. Reverse streamtracing allows more control over the grid size and a better resolution of final result.